We consider an anisotropic hyperbolic equation with memory term: $$ \partial_t^2 u(x,t) = \sum_{i,j=1}^n \partial_i(a_{ij}(x)\partial_ju) + \int^t_0 \sum_{| α| \le 2} b_α(x,t,η)\partial_x^αu(x,η) dη+ F(x,t) $$ for $x \in Ω$ and $t\in (0,T)$ or $\in (-T,T)$, which is a model equation for viscoelasticity. First we establish a Carleman estimate for this equation with overdetermining boundary data on a suitable lateral subboundary $Γ\times (-T,T)$. Second we apply the Carleman estimate to establish a both-sided estimate of $| u(\cdot,0)|_{H^3(Ω)}$ by $\partial_νu|_{Γ\times (0,T)}$ under the assumption that $\partial_tu(\cdot,0) = 0$ and $T>0$ is sufficiently large, $Γ\subset \partialΩ$ satisfies some geometric condition. Such an estimate is a kind of observability inequality and related to the exact controllability. Finally we apply the Carleman estimate to an inverse source problem of determining a spatial varying factor in $F(x,t)$ and we establish a both-sided Lipschitz stability estimate.
On solutions of matrix-valued convolution equations, anisotropic fractional derivatives and their applications in linear and non-linear anisotropic viscoelasticity
Sharp three sphere inequality for perturbations of a product of two second order elliptic operators and stability for the Cauchy problem for the anisotropic plate equation
Stabilization of the critical and subcritical semilinear inhomogeneous and anisotropic elastic wave equation
{ \bf Abstract} \，\，证明了有界区域上具有局部分布阻尼的临界和亚临界半线性非齐次各向异性弹性波动方程的指数衰减性。一个新颖之处是给出了非均匀各向异性介质的可检验条件。另一个新颖之处是建立了一个框架来研究阻尼半线性非齐次各向异性弹性波方程的稳定性，这是很难应用Carleman估计处理。我们发展了半线性弹性波动方程的Morawetz估计和紧唯一性论证，证明了唯一延拓性、能观性不等式和稳定性结果.指出我们的证明不同于经典方法（见Dehman et al. \cite{ZYY11}，Joly et al. \cite{ZYY16} and Zuazua \cite{ZYY43}），后者对亚临界半线性波动方程是成功的，对临界半线性波动方程是失败的.
Time-Dependent Potential Recovery from Local Data on Conformally Transversally Anisotropic Manifolds
School of Mathematics and Statistics, Huazhong University of Science and Technology（华中科技大学数学与统计学院）；Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology（华中科技大学工程建模与科学计算湖北省重点实验室）
